Magnetomer Direction

From Theory of Measurements Wiki

Jump to: navigation, search
[edit]

Problem Setup

A pulsation magnetometer is a measurement device, which measures the strength and direction magnetic field of the earth. This measurement can be expressed with a time-series

\mathbf{m}(t) = \int \mathbf{H}(t-\tau) \mathbf{AB}(\tau)d\tau +\xi(t)

By \mathbf{B} we denote the magnetic field, \mathbf{m} is the measured magnetic field, \mathbf{H} is the impulse response, \mathbf{A} is the projection matrix and \xi\, the noise process.

The projection matrix \mathbf{A} is formed of a set of linearly independent vectors x_1,x_2,x_3\,, which span the \mathbf{R}^3. We shall give them as follows.

x_1 = \begin{bmatrix}1\\0\\0\end{bmatrix}, x_2 = \begin{bmatrix}\cos(\theta_1) \\ \sin(\theta_1) \\ 0 \end{bmatrix} x_3 = \begin{bmatrix}\cos(\theta_2)\sin(\phi) \\ \sin(\theta_2)\sin(\phi) \\ \cos(\phi) \end{bmatrix} Now let us denote the projection matrix by

\mathbf{A} = \begin{bmatrix} x_1^T \\ x_2^T \\ x_3^T \end{bmatrix}

Let us now consider how to extract the parameters \sigma=(\theta_1,\theta_2,\phi)^T\, from geomagnetic data. Let us suppose, that we know exactly the impulse response \mathbf{H} and also the statistical properties of the noise process \xi\,. Now, the only thing we do not know are the parameter triplet \sigma\, and the magnetic field \mathbf{B}(t)\,.

Let us suppose, that we make measurement at time instances t_i\,, i=1,2,\dots\,. At every time instant, there are three measurements. However, at every time instant all the three components of the magnetic field are also unknown. Therefore we have always more unknowns than measurements. This makes the problem unstable. There are two solutions, either to increase the measurements or to use some a priori information. Let us consider the increase of measurements. The absolute magnetic field is also available. This means, that one measures

m_2(t_i) = \mathbf{B}(t_i)^T\mathbf{B}(t_i) + \xi_2(t)

From this we will get one more measurement per every time instant t_i\,. This gives thus an observation model

\mathbf{m}(t_i) = \int \mathbf{H}(t_i-\tau) \mathbf{AB}(\tau)d\tau +\xi(t_i) m_2(t_i) = \mathbf{B}(t_i)^T\mathbf{B}(t_i) + \xi_2(t)

These two equations can be given in general as

m(t) = f(\sigma,\mathbf{B};t) +\xi(t)

Let us assume, that the noise process is Gaussian process and can be defined by expectation and covariance \Sigma\,. Then the latter equation defines a distribution

D(\sigma,\mathbf{B}) \propto \exp\left(-\frac{1}{2}(m(t)-f(\sigma,\mathbf{B};t))^T\Sigma^{-1}(m(t)-f(\sigma,\mathbf{B};t))^T\right)

In order to get some estimate of σ and \mathbf{B}, we need to minimize the argument

\min (m(t)-f(\sigma,\mathbf{B};t))^T\Sigma^{-1}(m(t)-f(\sigma,\mathbf{B};t))^T

[edit]

Numerical Simulation

In numerical simulation, we use Marquardt-Levenberg (ML) minimization algorithm, when minimizing the argument. The algorithm can be found for example in \cite{NumRec}. For simulation purposes, we choose,

\theta_1 = 0.2\pi/2, \theta_2 = 0.1\pi/2, \phi=0.3\pi/2 \,

In order to simulate the magnetic field, we use random number generator. The field is given as

\mathbf{B}(t) \sim N(0,\sigma^2)

where \sigma^2 = 0.01\,.

We shall skip the simulation procedure, because it is just a technical detail for a geophysicist. Therefore we shall procede directrly to the results. When the simulation has been done, the ML algorithm results for different parameters are

\hat \theta_1=0.31415926535898, \hat \theta_2=0.15707963267949, \hat \phi = 0.47123889803847

By using the hat, we denote the estimate of the parameters. If we consider the relative errors, we will get

||\theta_1-\hat \theta_1||/\theta_1=0.1766974823035\cdot 10^{-15}, ||\theta_2-\hat \theta_2||/\theta_2 = 0.74212942567482\cdot 10^{-14} ||\phi-\hat \phi||/\phi= 0.7067899292141\cdot 10^{-15}

Thus, we can conclude, that the numerical simulation works.

Retrieved from "http://kavaro.com/mediawiki/index.php/Magnetomer_Direction"
Personal tools